1. Modifying arithmetic practice to promote understanding of mathematical equivalence Nicole M. McNeil University of Notre Dame

2. Seemingly straightforward math problem Mathematical equivalence problems 3 + 5 = 4 + __ 3 + 5 = __ + 2 3 + 5 + 6 = 3 + __

3. Theoretical reasons  Good tools for testing general hypotheses about the nature of cognitive development  E.g., transitional knowledge states, self-explanation, etc. Practical reasons  Mathematical equivalence is a fundamental concept in algebra  Algebra has been identified as a “gatekeeper” Why we care about these problems

4. Most children in U.S. do not solve them correctly 16% % of children who solved problems correctly Study

5. Why don’t children solve them correctly? Some theories focus on what children lack  Domain-general logical structures  Mature working memory system  Proficiency with “basic” arithmetic facts Other theories focus on what children have  Mental set, strong representation, deep attractor state, entrenched knowledge, etc.  Knowledge constructed from early school experience w/ arithmetic operations

6. But isn’t arithmetic a building block? Knowledge of arithmetic should help, right? Children’s experience is too narrow  Procedures stressed w/ no reference to =  Limited range of math problem instances Children learn the regularities  Domain-general statistical learning mechanisms that pick up on consistent patterns in the environment 2 + 2 = __ 12 + 8

7. Overly narrow patterns Perceptual pattern  “Operations on left side” problem format Concept of equal sign  An operator (like + or -) that means “calculate the total” Strategy  Perform all given operations on all given numbers 3 + 4 + 5 = __

8. Overly narrow patterns Perceptual pattern  “Operations on left side” problem format Concept of equal sign  An operator (like + or -) that means “calculate the total” Strategy  Perform all given operations on all given numbers

9. Overly narrow patterns Perceptual pattern  “Operations on left side” problem format Concept of equal sign  An operator (like + or -) that means “calculate the total” Strategy  Perform all given operations on all given numbers 3 + 4 = 5 + __

10. “Operations on left side” problem format

13. Equal sign as operator Child participant video will be shown

14. Add all the numbers Child participant video will be shown

15. Recap 2 + 2 = __ 12 + 8 2 + 2 = __ 12 + 8 3 + 4 + 5 = 3 + __

16. Internalize narrow patterns Recap 2 + 2 = __ 12 + 8 2 + 2 = __ 12 + 8

17. Internalize narrow patterns Recap 2 + 2 = __ 12 + 8 2 + 2 = __ 12 + 8 add all the numbers ops go on left side = means “get the total” 2 + 7 = 6 + __

18. The account makes specific predictions Performance should decline between ages 7 and 9 Traditional practice with arithmetic hinders performance Modified arithmetic practice will help

19. The account makes specific predictions Performance should decline between ages 7 and 9 Traditional practice with arithmetic hinders performance Modified arithmetic practice will help

20. Performance should get worse from 7 to 9 Why?  Continue gaining narrow practice w/ arithmetic  Strengthening representations that hinder performance But…  Constructing increasingly sophisticated logical structures  General improvements in working memory  Proficiency with basic arithmetic facts increases

21. Performance as a function of age Age (years;months) Percentage of children who solved correctly

22. The account makes specific predictions Performance should decline between ages 7 and 9 Traditional practice with arithmetic hinders performance Modified arithmetic practice will help

23. The account makes specific predictions Performance should decline between ages 7 and 9 Traditional practice with arithmetic hinders performance Modified arithmetic practice will help

24. Traditional practice with arithmetic should hurt Why?  Activates representations of operational patterns But…  Decomposition Thesis  “Back to basics” movement  Practice should “free up” cognitive resources for higher- order problem solving

25. 3 + 4 + 5 = 3 + __ SetReadySolve

26. Performance by practice condition Percentage of undergrads who solved correctly Practice condition

27. Performance should decline between ages 7 and 9 Traditional practice with arithmetic hinders performance Modified arithmetic practice will help The account makes specific predictions

28. Performance should decline between ages 7 and 9 Traditional practice with arithmetic hinders performance Modified arithmetic practice will help The account makes specific predictions

29. Performance by elementary math country Percentage of undergrads who solved correctly Elementary math country

30. Interview data Experience in the United States Experience in high-achieving countries 1 + 1 = 2 1 + 2 = 3 1 + 3 = 4 … 2 + 1 = 3 2 + 2 = 4 2 + 3 = 5 … 9 + 1 = 10 9 + 2 = 11 9 + 3 = 12 … 1 + 3 = 4 4 = 1 + 3 2 + 2 = 4 2 + 4 = 6 6 = 2 + 4 6 = 1 + 5 9 + 3 = 12 12 = 9 + 3 8 + 4 = 12

31. Effect of problem format Participants  7- and 8-year-old children (M age = 8 yrs, 0 mos; N = 90) Design  Posttest-only randomized experiment (plus follow up) Basic procedure  Practice arithmetic in one-on-one sessions with “tutor”  Complete assessments (math equivalence and computation)

32. Smack it (traditional format) 9 + 4 = __7 + 8 = __ 2 + 2 = __4 + 3 = __

33. Smack it (traditional format) 7 9 + 4 = __7 + 8 = __ 2 + 2 = __4 + 3 = __

34. Smack it (nontraditional format) 7 __ = 9 + 4__ = 7 + 8 __ = 2 + 2__ = 4 + 3

35. Snakey Math (traditional format)

36. Snakey Math (nontraditional format)

37. Understanding of mathematical equivalence  Reconstruct math equivalence problems after viewing (5 sec)  Define the equal sign  Solve and explain math equivalence problems Computational fluency  Math computation section of ITBS  Single-digit addition facts (reaction time and strategy) Follow up  Solve and explain math equivalence problems (with tutelage) Assessments

38. Summary of sessions Week 1Week 2Week 3Weeks 4-6 Traditional format Practice Session 1 Practice Session 2 10 min practice Assessments Follow up Nontraditional format Practice Session 1 Practice Session 2 10 min practice Assessments Follow up ControlAssessmentsPractice Sessions homework

39. Understanding of math equivalence by condition Arithmetic practice condition

40. Follow-up performance by condition Arithmetic practice condition

41. Computational fluency by condition MeasureControlTraditionalNontraditional Accuracy % correct (SD)86 (26)90 (25)92 (14) Reaction time M (SD)9.16 (6.80)6.98 (3.86)7.64 (4.08) ITBS score M NCE (SD)52.65 (20.14)53.00 (20.35)53.32 (18.08)

42. Computational fluency by condition MeasureControlTraditionalNontraditional Accuracy % correct (SD)86 (26)90 (25)92 (14) Reaction time M (SD)9.16 (6.80)6.98 (3.86)7.64 (4.08) ITBS score M NCE (SD)52.65 (20.14)53.00 (20.35)53.32 (18.08)

43. Interview data Experience in the United States Experience in high-achieving countries 1 + 1 = 2 1 + 2 = 3 1 + 3 = 4 … 2 + 1 = 3 2 + 2 = 4 2 + 3 = 5 … 9 + 1 = 10 9 + 2 = 11 9 + 3 = 12 … 1 + 3 = 4 4 = 1 + 3 2 + 2 = 4 2 + 4 = 6 6 = 2 + 4 6 = 1 + 5 9 + 3 = 12 12 = 9 + 3 8 + 4 = 12

44. Effect of problem grouping/sequence Participants  7- and 8-year-old children (N = 104) Design  Posttest-only randomized experiment (plus follow up) Basic procedure  Practice arithmetic in one-on-one sessions with “tutor”  Complete assessments (math equivalence and computation)

45. 4 + 6 = __ 4 + 5 = __ Traditional grouping 4 + 4 = __ 4 + 3 = __ In this example: 4 + n

46. 6 + 4 = __ 5 + 5 = __ Nontraditional grouping 4 + 6 = __ 3 + 7 = __ In this example: sum is equal to 10

47. Understanding of math equivalence by condition Arithmetic practice condition

48. Follow-up performance by condition Arithmetic practice condition

49. Computational fluency by condition MeasureControlTraditionalNontraditional Accuracy % correct (SD)94 (10)94 (11)98 (6) Reaction time M (SD)5.30 (2.60)5.56 (2.59)4.30 (1.56) ITBS score M NCE (SD)33.26 (14.22)50.35 (17.69)50.86 (13.49)

50. Computational fluency by condition MeasureControlTraditionalNontraditional Accuracy % correct (SD)94 (10)94 (11)98 (6) Reaction time M (SD)5.30 (2.60)5.56 (2.59)4.30 (1.56) ITBS score M NCE (SD)33.26 (14.22)50.35 (17.69)50.86 (13.49)

51. Performance declines between ages 7 and 9 Traditional practice with arithmetic hinders performance Modified arithmetic practice helps Summary

52. Implications Theoretical  Misconceptions not always due to something children lack  Limits of Decomposition Thesis  Learning may not spur conceptual reorganization Practical  Early math shouldn’t be dominated by traditional arithmetic  May be able to facilitate transition from arithmetic to algebra by modifying early arithmetic practice

53. Special thanks Institute of Education Sciences (IES) Grant R305B070297 Members of the Cognition Learning and Development Lab at the University of Notre Dame Martha Alibali and the Cognitive Development & Communication Lab at the University of Wisconsin Administrators, teachers, parents, and students Curry K. Software (helped us adapt Snakey Math)

54. 2 + 2  4 + 8

55. What other types of input might matter?